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# 机器学习代考_Machine Learning代考_CITS5508 Monte Carlo Reinforcement Learning

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## 机器学习代考_Machine Learning代考_Monte Carlo Reinforcement Learning

In model-free learning, the first problem faced by policy iteration algorithms is that policies become unevaluable since we cannot apply the law of total probability without knowledge about the model. As a result, the agent has to try actions and observe state transitions and rewards. Inspired by $K$-armed bandit, a straightforward replacement of policy evaluation is to approximate the expected cumulative rewards by averaging over the cumulative rewards of multiple samplings, and such an approach is called Monte Carlo Reinforcement Learning (MCRL). Since the number of samplings must be finite, the method is more suitable for reinforcement learning problems with $T$-step cumulative rewards.

The other difficulty is that policy iteration algorithms only estimate the value function $V$, but the final policy is obtained from the state-action value function $Q$. Converting $V$ to $Q$ is easy when the model is known, but it can be difficult when the model is unknown. Therefore, the target of our estimation is no longer $V$ but $Q$, that is, estimating the value function for every state-action pair.

Besides, when the model is unknown, the agent can only start from the initial state (or initial state set) to explore the environment, and hence policy iteration algorithms are not applicable since they need to estimate every individual state. For example, the exploration of watermelon planting can only start from sowing but not other states. As a result, we can only gradually discover different states during the exploration and estimate the value functions of state-action pairs.

Putting them all together, when the model is unknown, we start from the initial state and take a policy for sampling. That is, by executing the policy for $T$ steps, we obtain a trajectory
$$\left\langle x_0, a_0, r_1, x_1, a_1, r_2, \ldots, x_{T-1}, a_{T-1}, r_T, x_T\right\rangle$$

## 机器学习代考_Machine Learning代考_Temporal Difference Learning

When the model is unknown, Monte Carlo reinforcement learning algorithms use trajectory sampling to overcome the difficulty in policy evaluation. Such algorithms update value functions after each trajectory sampling. In contrast, the dynamic programming-based policy iteration and value iteration algorithms update value functions after every step of policy execution. Comparing these two approaches, we see that Monte Carlo reinforcement learning algorithms are far less efficient, mainly because they do not take advantage of the MDP structure. We now introduce Temporal Difference (TD) learning, which enables efficient model-free learning by joining the ideas of dynamic programming and Monte Carlo methods.

Essentially, Monte Carlo reinforcement learning algorithms approximate the expected cumulative rewards by taking the average across different trials. The averaging operation is in batch mode, which means state-action pairs are updated together after sampling an entire trajectory. To improve efficiency, we can make this updating process incremental. For state-action pair $(x, a)$, suppose we have estimated the value function $Q_t^\pi(x, a)=\frac{1}{t} \sum_{i=1}^T r_i$ based on the $t$ state-action samples, then, similar to (16.3), after we obtained the $(t+1)$-th sample $r_{t+1}$, we have
$$Q_{t+1}^\pi(x, a)=Q_t^\pi(x, a)+\frac{1}{t+1}\left(r_{t+1}-Q_t^\pi(x, a)\right),$$
which increments $Q_t^\pi(x, a)$ by $\frac{1}{t+1}\left(r_{t+1}-Q_t^\pi(x, a)\right)$. More generally, by replacing $\frac{1}{t+1}$ with coefficient $\alpha_{t+1}$, we can write the increment as $\alpha_{t+1}\left(r_{t+1}-Q_t^\pi(x, a)\right)$. In practice, we often set $\alpha_t$ to a small positive value $\alpha$. If we expand $Q_t^\pi(x, a)$ to the sum of step-wise cumulative rewards, then the sum of the coefficients is 1 , that is, letting $\alpha_t=\alpha$ does not change the fact that $Q_t$ is the sum of cumulative rewards. The larger the step-size $\alpha$ is, the more important the later cumulative rewards are.

## 机器学习代考Machine Learning代考_Temporal Difference Learning

$$Q_{t+1}^\pi(x, a)=Q_t^\pi(x, a)+\frac{1}{t+1}\left(r_{t+1}-Q_t^\pi(x, a)\right),$$

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